If for some positive integer $mathrm{n}$, the coefficients of three consecutive terms in the binomial
If for some positive integer $\mathrm{n}$, the coefficients of three consecutive terms in the binomial expansion of $(1+x)^{n+5}$ are in the ratio $5: 10: 14$, then the largest coefficient in this expansion is :-
Correct Option: , 3
Let $\mathrm{n}+5=\mathrm{N}$
$\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}}}: \mathrm{N}_{\mathrm{C}_{\mathrm{r}+1}}=5: 10: 14$
$\Rightarrow \frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}}}}{\mathrm{N}_{\mathrm{C}_{\mathrm{r}-1}}}=\frac{\mathrm{N}+1-\mathrm{r}}{\mathrm{r}}=2$
$\frac{\mathrm{N}_{\mathrm{C}_{\mathrm{r}+1}}}{\mathrm{~N}_{\mathrm{C}_{\mathrm{t}}}}=\frac{\mathrm{N}-\mathrm{r}}{\mathrm{r}+1}=\frac{7}{5}$
$\Rightarrow \quad r=4, N=11$
$\Rightarrow(1+x)^{11}$
Largest coefficient $={ }^{11} \mathrm{C}_{6}=462$